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**1 - 1**of**1**### From the Group SL(2, C) to Gyrogroups and Gyrovector Spaces and Huperbolic Geometry

"... this paper is to present a natural way in which the algebra of the SL(2; C) group leads to gyrogroups and gyrovector spaces. This natural way convincingly demonstrates that the theory of gyrogroups and gyrovector spaces provides a most powerful formalism for dealing with the Lorentz group and hyper ..."

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this paper is to present a natural way in which the algebra of the SL(2; C) group leads to gyrogroups and gyrovector spaces. This natural way convincingly demonstrates that the theory of gyrogroups and gyrovector spaces provides a most powerful formalism for dealing with the Lorentz group and hyperbolic geometry, the geometry that governs the special theory of relativity as well as other areas of physics (see, for instance, [9] and [10]). It is therefore hoped that, following this article, gyrogroup and gyrovector space theoretic techniques will provide standard tools in the study of relativity physics and, as such, will become part of the lore learned by all explorers who are interested in relativity physics. Links between gyrogroups and other mathematical objects are presented in [11] [12] [13] and [14]. Furthermore, our approach to gyrogroups and scalar multiplication in a gyrogroup of gyrovectors can be used as a preparation for the study of a related, but more abstract study of Sabinin's odules in [15]. A related study of quasigroups in differential geometry is presented by Sabinin and Miheev on pp. 357 -- 430 of [16]. 6 2 THE ALGEBRA OF THE SL(2; C) GROUP Let R 3 c be the set of all relativistically admissible velocities, R 3 c = fv 2 R 3 : kvk < cg It is the ball of radius c, c > 0, of the Euclidean 3-space R 3 , c being the vacuum speed of light. A boost L(v) is a pure Lorentz transformation, that is, a Lorentz transformation without rotation, parametrized by a velocity parameter v 2 R 3 c . The boost L(v) is a linear transformation of spacetime coordinates which has the matrix representation Lm (v), v = (v 1 ; v 2 ; v 3 ) t , Lm (v) = 0 B B B B B B @ v c 2 v v 1 c 2 v v 2 c 2 v v 3 v v 1 1 + c 2 2 v v +1 v 2 1 c ...